Question

If \( ^9C_3 + ^9C_5 = ^{10}C_r \) for some \( r \in \mathbb{N} \), then \( r = \) ?

• A. 3
• B. 4
• C. 5
• D. 7

Hint:

Use symmetry of combinations: \( ^nC_r = ^nC_{n-r} \), and Pascal's identity to simplify expressions involving sums of binomial coefficients.

Answer 1

The Correct Option is B

We are given: \[ ^9C_3 + ^9C_5 = ^{10}C_r \] Use the identity: \[ ^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1} \] Note that \( ^9C_3 + ^9C_5 \) doesn't directly follow the identity, but we can observe: \[ ^9C_3 = ^9C_6,\quad ^9C_5 = ^9C_4 \quad \text{(by symmetry)} \Rightarrow ^9C_6 + ^9C_4 = ^{10}C_4 \] Hence, \[ r = 4 \]